What Is the Exterior Angle Theorem?

The exterior angle theorem states that an angle exterior to a triangle equals the sum of the two angles not adjacent to it. The theorem’s basis derives from the fact that the sum of all the interior angles in a triangle equals 180 degrees.

The exterior angle theorem comes from Euclid’s “Elements,” Book 1, Proposition 16. The formal proof for the theorem uses the triangle sum theorem. For a triangle ABC with interior angles a, b and c, it follows that the sum of these angles is 180 degrees. Extending a side of the triangle at any given vertex creates an exterior angle d. Using the supplementary-angle theorem, which states that the sum of two adjacent angles formed by the intersection of two lines equals 180 degrees, the sum of angle d and its adjacent angle must equal 180 degrees. Thus, the sum of the remaining two adjacent angles must equal the angle d.

The exterior angle theorem helps determine unknown angles in a triangle so long as one of the interior angles is known. For triangle ABC and interior angles a, b and c, suppose angle c is known to be 35 degrees. Using the exterior angle theorem, angle d equals 145 degrees, so the interior angles a+b also equal 145 degrees. Further work can determine the value of the a and b angles.